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Increasing GP with AMs 6 and 54: Find AM of a, d | JEE

JEE Maths question with a full step-by-step solution.

Question

a, b, c, d

are in increasing G.P. If the A.M. between

a

and

b

6

and the A.M. between

c

and

d

54

, then the A.M. of

a

and

d

15

48

44

42

correct

Solution

Step 1: Let the ratio be

r

, so

b=ar,\ c=ar^2,\ d=ar^3

. The two given A.M.s become

\frac{a+ar}{2}=6\ \Rightarrow\ a(1+r)=12,\qquad \frac{ar^2+ar^3}{2}=54\ \Rightarrow\ ar^2(1+r)=108.

Step 2: Divide the second by the first:

\frac{ar^2(1+r)}{a(1+r)}=\frac{108}{12}\ \Rightarrow\ r^2=9\ \Rightarrow\ r=3\quad(r=-3\text{ rejected, G.P. is increasing}).

Step 3: Then

a(1+3)=12\Rightarrow a=3

, and

d=ar^3=3\cdot 27=81

. Step 4:

\frac{a+d}{2}=\frac{3+81}{2}=42.

Correct answer: (4)

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